Why did Gandalf choose Bilbo?

[Legate of Amon Lanc]

It's elaborated on quite a bit in "The Quest of Erebor" in the Unfinished Tales. Basically the point was that Gandalf had remembered him already from before, as one being curious about the adventurous stories and wide world, and after many years overhearing him being talked about by the neighbours as a bit queer one, talking to Dwarves and strangers and, very importantly, not being attached in any way (not having a family etc).

Suddenly in my mind these three things came together: the great Dragon with his lust, and his keen hearing and scent; the sturdy heavy-booted Dwarves with their old burning grudge; and the quick, soft-footed Hobbit, sick at heart (I guessed) for a sight of the wide world.